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In this article, we introduce the space $D([0,1];D)$ of functions defined on$[0,1]$ with values in the Skorohod space $D$, which are right-continuous andhave left limits with respect to the $J_1$ topology. This space is equippedwith the Skorohod-type distance introduced in Whitt (1980). Following theclassical approach of Billingsley (1968, 1999), we give several criteria fortightness of probability measures on this space, by characterizing therelatively compact subsets of this space. In particular, one of these criteriahas been used in the recent article Balan and Saidani (2018) for proving theexistence of a $D$-valued $\alpha$-stable L\'evy motion. Finally, we give acriterion for weak convergence of random elements in $D([0,1];D)$, and acriterion for the existence of a process with sample paths in $D([0,1];D)$based on its finite-dimensional distributions.

Weak convergence and tightness of probability measures in an abstract Skorohod space